Conditional probability

class: center, middle, title-slide, inverse
layout: false

# Conditional Probability
## Professor Andy Field

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???

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---
# Confusing conditional probabilities

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Vacinated </th>
   <th style="text-align:center;"> Covid </th>
   <th style="text-align:center;"> No Covid </th>
   <th style="text-align:center;"> Total </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Vaxed </td>
   <td style="text-align:center;"> 7 </td>
   <td style="text-align:center;"> 2993 </td>
   <td style="text-align:center;"> 3000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Not Vaxed </td>
   <td style="text-align:center;"> 293 </td>
   <td style="text-align:center;"> 3707 </td>
   <td style="text-align:center;"> 4000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Total </td>
   <td style="text-align:center;"> 300 </td>
   <td style="text-align:center;"> 6700 </td>
   <td style="text-align:center;"> 7000 </td>
  </tr>
</tbody>
</table>

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}) = \frac{300}{7000} = 0.043
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{vaxed}) = \frac{3000}{7000} = 0.43
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{vaxed}|\text{covid}) = \frac{p(\text{vaxed} \cap \text{covid})}{p(\text{covid})} = \frac{\frac{7}{7000}}{0.043} = 0.02
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}|\text{vaxed}) = \frac{p(\text{vaxed} \cap \text{covid})}{p(\text{vaxed})} = \frac{\frac{7}{7000}}{0.43} = 0.002
\end{aligned}`
$$
]
]

???

Data based on Indiana cases Jan-August 2021. All values divided by 1000 (so there's ~7 million residents of Indiana) and values rounded to make the maths simple.

Let's look at what happens as we increase vaccination rates - let's double them (so 6000 vaccinated people) and assume the rate of covid remains the same (so 7 increases to 14)

---

# Confusing conditional probabilities

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Vacinated </th>
   <th style="text-align:center;"> Covid </th>
   <th style="text-align:center;"> No Covid </th>
   <th style="text-align:center;"> Total </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: yellow !important;"> Vaxed </td>
   <td style="text-align:center;background-color: yellow !important;"> 14 </td>
   <td style="text-align:center;background-color: yellow !important;"> 5986 </td>
   <td style="text-align:center;background-color: yellow !important;"> 6000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Not Vaxed </td>
   <td style="text-align:center;"> 286 </td>
   <td style="text-align:center;"> 714 </td>
   <td style="text-align:center;"> 1000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Total </td>
   <td style="text-align:center;"> 300 </td>
   <td style="text-align:center;"> 6700 </td>
   <td style="text-align:center;"> 7000 </td>
  </tr>
</tbody>
</table>

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}) = \frac{300}{7000} = 0.043
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{vaxed}) = \frac{6000}{7000} = 0.86
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{vaxed}|\text{covid}) = \frac{p(\text{vaxed} \cap \text{covid})}{p(\text{covid})} = \frac{\frac{14}{7000}}{0.043} = 0.05
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}|\text{vaxed}) = \frac{p(\text{vaxed} \cap \text{covid})}{p(\text{vaxed})} = \frac{\frac{14}{7000}}{0.43} = 0.002
\end{aligned}`
$$
]
]

???

As vaccinations increase, p(vaxed|covid) increases (it doubles because there are more vaccinated people and therefore more vaccinated covid cases). This is the conditional probability that antivaxxers focus on "Oh, the vaccination is making it worse because the number of people with covid who are vaxed has increased. However, this isn't what we want to know, we want to know the probability of having covid given you;'re vaxxed. This is unchanged by increasing numbers of people becomming vaccinated.

---
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Vacinated </th>
   <th style="text-align:center;"> Covid </th>
   <th style="text-align:center;"> No Covid </th>
   <th style="text-align:center;"> Total </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Vaxed </td>
   <td style="text-align:center;"> 7 </td>
   <td style="text-align:center;"> 2993 </td>
   <td style="text-align:center;"> 3000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Not Vaxed </td>
   <td style="text-align:center;"> 293 </td>
   <td style="text-align:center;"> 3707 </td>
   <td style="text-align:center;"> 4000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Total </td>
   <td style="text-align:center;"> 300 </td>
   <td style="text-align:center;"> 6700 </td>
   <td style="text-align:center;"> 7000 </td>
  </tr>
</tbody>
</table>

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{not vaxed}) = \frac{4000}{7000} = \frac{4}{7} = 0.57
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}|\text{vaxed}) = \frac{p(\text{vaxed} \cap \text{covid})}{p(\text{vaxed})} = \frac{\frac{7}{7000}}{0.43} = 0.002
\end{aligned}`
$$
]
]

.ong[
.eq_med[
$$
`\begin{aligned}
p(\text{covid}|\text{not vaxed}) = \frac{p(\text{not vaxed} \cap \text{covid})}{p(\text{not vaxed})} = \frac{\frac{293}{7000}}{0.57} = 0.07
\end{aligned}`
$$
]
]

.ong[
.eq_lrge[
$$
`\begin{aligned}
\frac{p(\text{covid}|\text{not vaxed})}{p(\text{covid}|\text{vaxed})} = \frac{0.07}{0.002} = 35
\end{aligned}`
$$
]
]

???

The probability of having covid if unvaxed is 35 times higher than that if you are vaxed